The game starts with no monarch. Once an effect makes one player the monarch, the game will have exactly one monarch from that point forward.

8/23/2016

Abilities that trigger whenever you “become the monarch” trigger only if you aren’t already the monarch. For example, if you are already the monarch as Custodi Lich enters the battlefield, its last ability won’t trigger.

8/23/2016

If the triggered ability that causes the monarch to draw a card goes on the stack, and a different player becomes the monarch before that ability resolves, the first player will still draw the card.